Probabilități

1.Legea normala 1dimensionala:

x-v.a. cu fct de rep F(x)=> [a,b,)=(-µ,b](µ,a) => P(XÎ[A,B))=P(X<B)-P(X<a) f densitatea lui x => P(X<b)-P(x<a)=ò(-µ,b)f(x)dx-ò(-µ,a)f(x)dx=ò(a,b)f(x)dx

P(XÎ[a,b])=P(xÎ[a,b]+P(x=b)=P(XÎ[a,b]] deorece P(x=b)=0=>P(XÎ[a,b]=ò(a,b)f(x)dx

Transformarea liniara de forma Y=sx+n s,nÎR,s>0=> y<x óx<(x-m)/s => Y are o fct de repartitie Fy(x)=F((x-m)/s); P(YÎ[a,b])=F((b-m)/s)-F((a-m)/s); P(YÎ[a,b])=F((b-m)/s-F((a-m)/s); Fy(x)=ò(-µ,(x-m)/s)f(t)dt=ò(-µ,x)1/sf((y-m)/s)dy; =>Y are densitatea fy(x)=1/sf((x-n)/s)

Def

X-v.a cu rep normala standard dk are densitatea f(x)=1/Ö2pe^(-(x^2)/2)=> E[X]=0, V[X]=1 v.a. Normala cu medie m si varianta s^2 o v.a. de forma Y=sX+m;

x-v normala standard s,mÎR,s>0=>Y~N(m,s);

Y are densitatea f(x)=1/Ö2pe^(-(x-m)^2/(2s^2));

Functia de rep alui X~N(0,1) se noteaza F(x); F(x)=1/Ö2pò(-µ,x)e^(-t^2/2)dt;

F(-x)=1-F(x);

Y~N(m,s)=> P(YÎ[a,b])=F((b-n)/s)-F((a-m)/s); in part P(|Y-m)<=3s)=P(YÎ[m-3, m+3s])=F(3)-F(-3)=2F(3)-1~=0,997 Legea 3s Cu probab 0,997, abaterea fata de medie este <=3s

2. Mat d corel & mat d cov a unui v.a

Def X=[x1…xn]vert-vect aleator n dimensional

Matricea de corel: Rx=[E[XiXj]] unde E[XiXj] media produsului XiXj

Matricea de cov Kx=[cov(xi,xj)]

Obs:1. Cov(xi,xj)=E(Xi’Xj’] media prod var centrate xi’=xi-E[xi];xj’=xj-E[xj] ; daca x’=[x1’…xn’]vert Kx=Rx’;

2. Cele doua matrici Rk,Kx sunt simetrice, avand pe diag princip E[x1^2]..E[xn^2] respectiv V(x1), …V(xn)

3. x1…xn doua cate doua independente=>necorelate=> cov(xi,xj)=0, i¹j=> Kx matrice diagonala

Prop(sch matricii de corelatie dupa transf liniara aplicata vect aleator X) X-vector aleator n-dimensional; Y=AX, A-matrice patratica => Ry=ARxA^t; Corolar (v) m=[m1..mn] ÎR^n, Kx a vect Y=AX+m este Ky=AKxA^t; Dem: Liniaritatea mediei=> E[X]=AE[X]+E[m]=AE[X]+m; y’=Y-E[Y]=AX+m-AE[x]-m=A[X]-AE[X]=AX’; Ry’(=Ky)=ARx’A^t=AKxA^t;

3.Distribuita normala n-dimensionala

Def: X=[x1..xn] v.a. n dim; X urmeaza o distrib normala standard daca x are densitatea de rep f(x1,..xn)=1/((Ö2p)^n)e^(-1/2(x1^2+..xn^2))

Obs

1. f(x1,..xn)=f1(x1),f2(x2)..fn(n), cu fi(xi)=1/((Ö(2p))e^(-1/2 xi^2) => x1,…xn doua cate doua independente si urmeaza fiecare o distributie normala standard 1 dimensionala (xi~N(0,1))

2. 2. Inlocuind x1^2+..Xn^2 cu xx=x^tx=> f(x)=1/(Ö(2p)^n)e(-1/2x^tn)

Def : Va n-dim Y urmeaza o distributie normala daca este de forma Y=AX+n; A-matrice patratica nesingulara; m-vector din Rn; X=v.a n-dimensional cu distrib normala standard